# Does an audio equalizer consist of a perfect-reconstruction filterbank?

Within the theory of perfect reconstruction, I discovered that it is not obvious that the combination of all filters in a filterbank gives you the original signal. Therefore, I was wondering whether an audio equalizer (either analog or digital) distorts the spectrum of a signal, even when the equalizer has 0 dB gain in each frequency band. Is this ever considered a problem?

• SE.DSP wishes you a happy new year, with a reminder that your question and its answers may require some action (votes, acceptance, etc.) – Laurent Duval Dec 31 '16 at 16:22

A filter bank system is generally composed of:

• an analysis filter bank, that split an input signal into components with filters and may reduce their rate
• a synthesis filter bank (FB), that takes the components, potentially increase their rate, and feed them into filter elements

so that the novel outputs can be combined into a signal. It is perfect reconstruction (PR) when, for every input, the output is a potentially delayed or scaled version of the input. Filters can be nonlinear, the rate reduction can be different for each component. Min-Max or morphological wavelets are examples of the above. There are even hybrid filter banks, where the analysis (A) or synthesis (S) FB are either analog or digital. Their design is quite involved, so often A and S are linear, with upsampling and downsampling operators, and an output obtained by summation, as explained by @Marcus Müller. However, the filters may have different bandwidths and the subsampling can be different for each filter.

Many signal processing systems, such as (most) equalizers, are made of an analysis FB, a processing unit (often nonlinear) and a synthesis FB (which could be a mere summation). The non-perfection arises, for PR FB, because of the middle filtering component.

Assuming an equalizer has linear A and S filters, the non PR aspect may arise from two aspects. First, what are the filters? It seems to me that the traditional filters (first or second order) do not allow easy PR. However, with sufficient frequency overlap, they together can form a near PR filter banks, with an error below the hearing threshold. The second aspect is the processing. If you suspect saturation, or nonlinear phase alteration, then PR is not achieved in general.

Phase distortion is sometimes considered harmless for audio in some textbooks. However, picky sound engineers are cautious about audible phase distortion. The article Plugging into EQ discusses those issue, around FIR or IIR filters. The first ones can have linear phase, but have less sharp frequency transitions, and require a handful of computations. The second ones do not have linear phase in general. It also provides a comparative evaluation of several EQ.

So, as Markus, I'll say no in general, with a modest yes (in cases) if below the hearing sensitivity.

• In an qualizer the processing unit would be linear. – Olli Niemitalo Oct 7 '16 at 19:08
• Never a saturation? – Laurent Duval Oct 7 '16 at 19:10
• @OlliNiemitalo Does any equalizer leave a constant input unharmed? – Laurent Duval Oct 7 '16 at 19:15
• If you really drive a hardware equalizer there could be saturation, but it is not the way it should be operated. VST effect plugins use floating point. Audio equipment often have AC coupled input so that it can be biased to an average DC voltage that is in the middle of the voltage range that the electronics can handle, but that is not needed with pure software. – Olli Niemitalo Oct 7 '16 at 19:25

Does an audio equalizer consist of a perfect-reconstruction filterbank?

Short answer: No. It's a bit unsharp what you're asking here, since reconstruction filterbanks are usually things that combine multiple, separate, independent signals back into one signal, but let's assume this makes sense here, and your audio source magically produces one signal stream per equalizer filter (it doesn't).

The perfect-reconstruction criteria typically applied to filterbanks is the following [1, p. 288]:

\begin{align} \tag{10.47} \frac1N \sum\limits_{i=0}^{N-1} G_i(Z^N)H_i(Z^N) &= Z^{-K} \\ \tag{10.48} \sum\limits_{i=0}^{N-1} G_i(Z^H)H_i(Z^N) e^{-j\frac{2\pi}{N}im} &=0;\quad1\le m \le N-1\\ \text{with}\\ G_i&\quad\text{the reconstruction filterbank}\\ H_i&\quad\text{the decomposition filterbank} \end{align}

As you can see, this equation has two aspects:

1. first of all, $(10.47)$ dictates that result of passing the signal through both the decomposition filterbank (in that case: whatever frequency-selective behaviour your audio channel has) and then through the reconstruction filterbank (your equalizer). Generally, audio filters don't care about phase reconstruction as much – and hence, that can't be achieved. However, it could be achieved if you had an equalizer designed to do so, so let's consider this fulfilled.
2. the absence of aliasing between the results of the individual filters. Now look closely at the $e^{\cdot}$ term: it is exactly the coefficients of the $m^\text{th}$ row of the DFT matrix; in other words: the filterbank filters need to be equidistant in frequency, if they are of the same (yet shifted) spectral shape. In audio equalizers, however, the same shape, yet scaled is found, typically on logarithmic intervals in frequency. That can't work out, since you necessarily violate the orthogonality constraint $(10.48)$ for some $m$.

Is this ever considered a problem?

I'm not enough of an Audio application expert to tell, but I guess that, yes, everything is a problem as soon as your requirement (for example, phase continuity) gets harsh enough – what I could imagine happens easily for audio environments with a large delay spread.

[1] Maurice Bellanger, Digital Processing of Signals. Second Ed., John Wiley and Sons, 1988

• Can you explain intuitively why the filters HAVE to be equidistant and with the same shape? Imagine a simple fourier filter bank (which is perfectly reconstructable). If you add two channels into one, it doubles in width and the whole filterbank is still perfectly reconstructable, isn't it? – BNJMNDDNN Oct 7 '16 at 17:16
• @BNJMNDDNN isn't the $e^{-j\ldots}$ term in $(10.48)$ intuitive enough? – Marcus Müller Oct 7 '16 at 18:19
• Oke. so to my second question, the answer is "no"? – BNJMNDDNN Oct 12 '16 at 11:57

It's possible for the result of an audio equalizer bank (especially an analog one) set for flat gain to be equivalent to an all-pass filter, which would not be a perfect reconstruction process due to potential non-linear phase shifts.